The Stacks project

Disjoint unions of sheaves are open and closed

Thanks to Laurent Moret-Bailly
https://stacks.math.columbia.edu/tag/02WN#comment-3519

LaTeX

Added a new macro

\def\Im{\text{Im}}

and replaced all occurrences of \text{Im} by \Im

Whitespace changes

LaTeX: \Ob

	Introduced a macro

	\def\Ob{\mathop{\rm Ob}\nolimits}

	and replaced any occurence of \text{Ob}( with \Ob(. There are
	still some occurences of \text{Ob} but these are sets, not the
	operator that takes the set of objects of a category.

LaTeX: \Sch

	Introduced a new macro

	\def\Sch{\textit{Sch}}

	and replaced all the occurences of \textit{Sch} with \Sch.

Descent: morphisms of schemes satisfy fpqc descent

	We edited the descent chapter to improve our exposition,
	prompted by a question of Thanos D. Papaïoannou, namely

	"Do morphisms of algebraic spaces satisfy fpqc descent?"

	Here is one answer:

	Suppose X, Y are algebraic spaces over an affine base S. Suppose
	that S' --> S is a surjective flat morphism of affine schemes.
	Finally suppose that a' : X_{S'} --> Y_{S'} is a morphism of
	algebraic spaces over S' which is compatible with the canonical
	descent data. Now you want to know if a' descends to a morphism
	a : X --> Y over S.

	If X, Y are representable, then this is Descent, Lemma Tag 02W0.
	See also the explanation in the remark following that lemma. The
	proof is kind of long since I tried to prove somehow more
	general statements in that section. The key is Descent Lemma Tag
	0241. It mainly relies on the fact that the representable
	presheaf associated to a scheme satisfies the sheaf condition
	for the fpqc topology (which is Descent, Lemma Tag 023Q). The
	analogue for this in the category of algebraic spaces is
	Properties of Spaces, Lemma Tag 03WB but there is a condition,
	namely that the algebraic space is Zariski locally quasi-separated.

	So I do not know how to prove this descent in general, and it
	may not be true. But it is true for Zariski locally
	quasi-separated spaces. Eventually we will state this descent
	property explicitly in the stacks project.

	Any algebraic space in the literature, say published before year
	2000, is quasi-separated (since there are basically no
	references which deal with more general ones). Hence fpqc
	descent for morphisms of algebraic spaces is true for any
	algebraic spaces which you can find in these articles/books.

Added new tags to stacks project

	modified:   tags/tags

More changes in spaces.tex

	modified:   fpqc-descent.tex
	modified:   spaces.tex

More changes in spaces.tex

	modified:   fpqc-descent.tex
	modified:   spaces.tex

Beginning work on algebraic spaces

	modified:   algebra.tex
	modified:   fpqc-descent.tex
	modified:   groupoids.tex
	modified:   morphisms.tex
	modified:   sites.tex
	modified:   spaces.tex